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D preserves real-rootedness by Rolle's theorem and p + c p' is real-rooted for every real c by Rolle's theorem applied to e^(x/c) p; Polya--Schur 1914); when the coefficients of phi have one sign, coefficients of one sign stay of one sign; applied to phi(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.splits_add_C_mul_derivative_all","RealRooted.splits_C_mul_add_derivative","Polynomial.Splits.derivative"],"certificate_hash":"ee6aba6c689e55364e57fa9470f624b051a4d0f68a2e2b9bca78c0127579f276","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-940e4d08e5526f20","role":"sequence","name":"reversal of Riordan array (1/((1-x)(1-3x)),x/((1-x)(1-3x))).","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:  1;\n  1:  5,    4;\n  2:  9,   34,    13;\n  3: 13,   96,   162,     40;\n  4: 17,  190,   639,    668,    121;\n  5: 21,  316,  1636,   3420,   2551,     364;\n  6: 25,  474,  3345,  10856,  16134,    9294,    1093;\n  7: 29,  664,  5958,  26560,  61490,   70128,   32804,    3280;\n  8: 33,  886,  9667,  55140, 176135,  313052,  287874,  113176,   9841;\n  9: 37, 1140, 14664, 102228, 419601, 1031388, 1476180, 1133016, 383817, 29524;\n","recurrence_sentence":"The row polynomials satisfy (16*n^3-20*n^2+6*n+(8*n^3+8*n^2-22*n)*x+(16*n^2-16*n)*x^2+8*n*x^3)*P(n,x) = (16*n^3-4*n^2-2*n+(104*n^3-68*n^2-58*n)*x+(80*n^3+40*n^2-184*n+40)*x^2+(16*n^3+112*n^2-116*n+28)*x^3+(32*n^2+16*n-16)*x^4+(16*n-16)*x^5)*P(n-1,x) - ((8*n^2+2*n)*x+(68*n^2+28*n-3)*x^2+(64*n^2+112*n-20)*x^3+(16*n^2+112*n+24)*x^4+(32*n+48)*x^5+16*x^6)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((16*n^3 - 4*n^2 - 2*n + (104*n^3 - 68*n^2 - 58*n)*x + (80*n^3 + 40*n^2 - 184*n + 40)*x^2 + (16*n^3 + 112*n^2 - 116*n + 28)*x^3 + (32*n^2 + 16*n - 16)*x^4 + (16*n - 16)*x^5)*P[n - 1, x] - ((8*n^2 + 2*n)*x + (68*n^2 + 28*n - 3)*x^2 + (64*n^2 + 112*n - 20)*x^3 + (16*n^2 + 112*n + 24)*x^4 + (32*n + 48)*x^5 + 16*x^6)*D[P[n - 1, x], x])/(16*n^3 - 20*n^2 + 6*n + (8*n^3 + 8*n^2 - 22*n)*x + (16*n^2 - 16*n)*x^2 + 8*n*x^3)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((16*n^3 - 4*n^2 - 2*n + (104*n^3 - 68*n^2 - 58*n)*x + (80*n^3 + 40*n^2 - 184*n + 40)*x^2 + (16*n^3 + 112*n^2 - 116*n + 28)*x^3 + (32*n^2 + 16*n - 16)*x^4 + (16*n - 16)*x^5)*v[i - 1] - ((8*n^2 + 2*n)*x + (68*n^2 + 28*n - 3)*x^2 + (64*n^2 + 112*n - 20)*x^3 + (16*n^2 + 112*n + 24)*x^4 + (32*n + 48)*x^5 + 16*x^6)*deriv(v[i - 1], x))/(16*n^3 - 20*n^2 + 6*n + (8*n^3 + 8*n^2 - 22*n)*x + (16*n^2 - 16*n)*x^2 + 8*n*x^3)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-6466a8550c08a52f","steps":["reverse","D+1"],"edges":["e-18c37bd14c5a739f","e-53f7dacbfa6d0f66"],"text":"A116414 \u2192 reverse \u2192 D+1"},"headline":"A116414 \u2192 reverse \u2192 D+1","chain":null}